Arrange a rack to the upper third position of oven. Preheat oven to broil.

Heat oil in a large skillet over high heat until shimmering, about 30 seconds. Add onions, toss until well coated in oil, and cook undisturbed until browned on one side and slightly softened, 5 to 7 minutes. Stir onions and continue to cook undisturbed until browned on other side, 5 to 7 minutes.

Add salt and beef broth, and cook while scraping up any brown bits stuck to the bottom of the pan until most of the liquid is absorbed, about 2 minutes. Stir in Worcestershire sauce, vinegar, thyme, and black pepper, and cook until fragrant and onions are brown all over, about 1 minute. Remove from heat.

Whisk together mayo and Dijon mustard until smooth; transfer Dijonnaise sauce to a squeeze bottle if desired.

Place hotdog buns on a large rimmed baking sheet. Add a hotdog to each bun, drizzle with prepared Dijonnaise sauce, and top with about 1/3 cup caramelized onions. Sprinkle each hotdog with a generous amount of cheese.